perhaps thats the reason, why it is so hard to believe, that the value of the number will be 1 when it comes to infinity...
Yes, but I think it's confusing properties of the irrationals with rationals.
e.g pi is equal to pi. It's the ratio between the circum and diam. of a circle.
If you were to say what is pi = a/b? Then you can't. You can get as close as dammit, but not equal.
Similary, if you say
3 < pi
3.1 < pi
3.14 < pi
and so on, you get as close as you like, but never there.
OTOH.
0.1111 rec equals 1/9. You're there already, there's no closer to get. Nothing you can add to get closer, no number you can express that is between them. Even though you can say
0.1 < 1/9 and
0.11 < 1/9
0.111 rec is there.
Similary, 0.999 rec is already one, you can't get closer [whereas you can trivially get closer if you have 0.999 or 0.9999999999 or 0.999<10 billion billion billion 9's>], it doesn't need to change into 1.
OTOH if you have a problem than 0.999 looks different, say
"one equals naught point nine nine recurring"
Translate that into 200 languages. Most of which you won't understand. Write the expression in multiple bases. Find alien life and see how they write the expression.
All of which may help lose that we're "taught" to do maths for years and years before we're taught the stuff behind it [indeed most of us don't get that far - plenty will tell you pi is 22/7 because of that]
i.e If it's the representation that's confusing, change it, but note that the proofs behind maths don't rely on the representation.
You're doing nothing different when you say 1/9 * 9 = 1. Except that when we learn how to do sums with fractions when we're primary school kids [well, it's probably A level stuff these days

] , we get a few tricks that perhaps make it easier to accept on face value that 9/9 = 1. But if you can accept the fraction then any, even philosophical sense, that you might feel 0.9999999999 rec isn't quite the same as 1 is in that fraction too.
e.g an axiom says that there's a rational number such that if you multiply it by 3 you get 1. 1/3 is a symbol for that number, so is 0.3333333 rec.
The other issue is that of what you can draw or write on bits of paper. Maths is abstract. Circles, rectangles and stuff like that don't actually exist, you can't draw them, does that mean you struggle to accept that a rectangle can have a specific area?
What about a piece of A4 paper? It isn't actually a rectangle. Similar a number like "4.24" you might be able to write on that piece of paper, but that's no better or worse a symbol than pi or 0.9999 rec, simply because it has some kind of finite length in one number system.
It has only very limited relationship with anything you can actually physically make or manipulate. 9/9 will never change into 1 either, unless you've very special paper.
Nevertheless there's a definite sense in which maths works [I can get my glass to fit in the frames when I treat them like rectangles] So I would say, in the abstract sense maths has things that are equal to other things [as well as close approximations and isomorphism and all that other jazz] but whenever you try to take even the most simple finite maths and say "I can do this on a piece of paper..." you'll be struggling to have anything that is exact or equal - I don't think you need an infinity of 9s to get to that issue though.
Or another way, say you and 2 friends did some work on my house and I said I'll split £369 quid with you, 1/3 each..keeping it nice and straightforward so we don't have to argue over splitting £1. You're happy that you get £123 each.
Then on the day I give you 0.33333 rec each and you each take the £123 but one of you says "Eh, I know a bit of maths, he's only given us 0.9999 rec, he's kept some of the money" and you beat me up for the rest of it. When the other one says "Hang on, we've got £369 between us and £123 each, his pockets are empty....." and the 3rd guy says "Ah! I asked my friend, he says it's because he only kept a very tiny amount" Principles being what they are you all beat each other up until the police arrive arguing over which pile of £123 had the tiny amount less
